The completion and Krull’s generalized principal ideal theorem on r-Noetherian rings

ABSTRACT

A ring is called an r-Noetherian ring if every regular ideal is finitely generated. Let R be an r-Noetherian ring, let I be a regular ideal of R, and let $\widehat{R}$ be the I-adic completion of R. We show that $\widehat{R}$ is a Noetherian ring and dim$\left(\widehat{R}\right)$  =  sup{r-ht(M)∣M∈Max(R) and I⊆M}. Let P be a prime ideal of R. We also prove that for any a∈reg(P), r-htP = ht(P∕aR)+1 and that if P is minimal over an n-generated regular ideal, then r-htP≤n.

KEYWORDS:

• Ideal-adic completion
• Krull dimension
• Krull’s generalized principal ideal theorem
• r-Noetherian ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13B35
• 13E05
• 13F25

Acknowledgment

The authors would like to thank the referee for his/her several valuable comments and suggestions.